Step |
Hyp |
Ref |
Expression |
1 |
|
madetsumid.a |
|- A = ( N Mat R ) |
2 |
|
madetsumid.b |
|- B = ( Base ` A ) |
3 |
|
madetsumid.u |
|- U = ( mulGrp ` R ) |
4 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
5 |
3 4
|
mgpbas |
|- ( Base ` R ) = ( Base ` U ) |
6 |
3
|
crngmgp |
|- ( R e. CRing -> U e. CMnd ) |
7 |
6
|
adantr |
|- ( ( R e. CRing /\ M e. B ) -> U e. CMnd ) |
8 |
1 2
|
matrcl |
|- ( M e. B -> ( N e. Fin /\ R e. _V ) ) |
9 |
8
|
adantl |
|- ( ( R e. CRing /\ M e. B ) -> ( N e. Fin /\ R e. _V ) ) |
10 |
9
|
simpld |
|- ( ( R e. CRing /\ M e. B ) -> N e. Fin ) |
11 |
|
simpr |
|- ( ( R e. CRing /\ M e. B ) -> M e. B ) |
12 |
1 4 2
|
matbas2i |
|- ( M e. B -> M e. ( ( Base ` R ) ^m ( N X. N ) ) ) |
13 |
|
elmapi |
|- ( M e. ( ( Base ` R ) ^m ( N X. N ) ) -> M : ( N X. N ) --> ( Base ` R ) ) |
14 |
11 12 13
|
3syl |
|- ( ( R e. CRing /\ M e. B ) -> M : ( N X. N ) --> ( Base ` R ) ) |
15 |
14
|
adantr |
|- ( ( ( R e. CRing /\ M e. B ) /\ r e. N ) -> M : ( N X. N ) --> ( Base ` R ) ) |
16 |
|
simpr |
|- ( ( ( R e. CRing /\ M e. B ) /\ r e. N ) -> r e. N ) |
17 |
15 16 16
|
fovrnd |
|- ( ( ( R e. CRing /\ M e. B ) /\ r e. N ) -> ( r M r ) e. ( Base ` R ) ) |
18 |
17
|
ralrimiva |
|- ( ( R e. CRing /\ M e. B ) -> A. r e. N ( r M r ) e. ( Base ` R ) ) |
19 |
5 7 10 18
|
gsummptcl |
|- ( ( R e. CRing /\ M e. B ) -> ( U gsum ( r e. N |-> ( r M r ) ) ) e. ( Base ` R ) ) |